Conservación y manejo del producto

This paragraph describes the modelling approach adopted in order to assess to what extent results obtained in the previous chapter are plagued by the presence of sample selection bias. Is it correct to focus the analysis of wage transitions only on those individuals for whom a valid wage is available in each panel wave? Or, on the contrary, are these individuals systematically different from the ones dropping out from the survey, so that their selection for estimation is endogenous, thus biasing estimation results?

To provide an answer to these questions, the (structure of the) model in Chapter 4 has been extended to allow for a third dichotomic event which interacts with the ones previously considered (i.e low-pay/high-pay at both ends of the transition investigated) in determining the likelihood of the data. The resulting set-up is a trivariate conditional probit model which allows for sequential nesting of the equation of interest.

The modelling of attrition is carried out expanding the model for low-pay persistence proposed by Stewart and Swaffield [1999], i.e., differently from the model of Chapter 4, the 1995 wage outcome is assumed to be observable only for the 1993 low-paid. The model is expanded by acknowledging that it can be actually estimated only using observations for which a wage is observable in 1993 and 1995. Let R| be a dummy partitioning the sample of 1993 wage earners77 depending upon their wage observability in 1995. Recalling the notation from Chapter 4, let us also define dit as dummy variables for low-pay occurrence in year t.

5. Discontinuous wage profiles, endogenous selection and mobility: a simulated estimation

approach

Recall from section 5.2. that these are wage earners belonging to panel households, this last state

At the first nesting level, a probit is estimated for the probability of having a valid wage in both periods.78 At the second nesting level, only those observations for which R|=1 are utilised to estimate a probit for the probability of low-pay in 1993. Finally, the sample of the 1993 low-paid observed in both waves is used to estimate a probit equation for low-pay in 1995. Of course, the probabilities of the three events above are simultaneously estimated, i.e. for those observation for which the 1995 wage outcome is observed the estimated probability is Prob(Ri=1,di93=1,di95=1)= Prob(dig5=1|di93=1,Rj=1)Prob(di93=1|Rl=1)Prob(Ri=1). It is worth stressing that the multivariate normal density assumed allows for unrestricted correlation between the errors, thus allowing a proper assessment of potential endogeneity issues among the three events investigated.

More formally, let us assume that the propensity to stay in sample (retention propensity) is a latent variable R*; when R* overcomes an unobservable (possibly individual specific) threshold x*. observations remain in the sample of wage earners in both waves. R* is assumed to be a function of observable characteristics, and we only observe a dummy indicator signalling whether or not R*>x*:

Rj = x'ro s r + vi

R, = / ( « ; > t?) (5.1)

v, ~ N(0,1)

where x'Ri contains the whole set of explanatory variables used In the model.

The second stage can be formalised according to the discussion in Chapter 4 and assuming partial observability of the 1993 low-pay outcome:

m Note that specification in terms of retention, rather than attrition, characterises the models both of Hausman and Wise [1979] and Bingley et al. [1995].

5. Discontinuous wage profiles, endogenous selection and mobility: a simulated estimation

approach

5. Discontinuous wage profiles, endogenous selection and mobility: a simulated estimation

approach

9(wm ) = x'i 8 + ui if R, = 1

d,93 = Hg{Wi93) * 9(^93)) (5.2)

ui ~ N( 0,1)

The headline equation of interest is a probit equation for the occurrence of low- pay in 1995 for which two sources of partial observability are assumed:

hi (w,95) = Z/'n-i + e1( if Rj = 1 and di93 = 1

^/95 = K^ t i wi95) — ^1(^95)) (5.3)

6! ~ A/(0,1)

Assuming that error terms in the three equations are jointly distributed as a tri­ variate normal

f v , ' O' r 1

Ui ~n3 0 _{P1 1}

-E/1- ,0, VP2 P3 t

and that observations are iid, the log-likelihood function of the model may be written

lo9(- = X , { ^ '9 3 ^ /9 5 SRtx’,- P,z',- yi;p i.P2.P3) +

Ri di93^ - c,/95)Io9 ‘I)3(x'R/ 8R.x'/ f t - * '/ ri;P l.-P 2 -P 3 ) +

Rii^ ~ di93) log4>2(X R/ 8r x'/ P;-Pi) +

(1 -R ( )lo g O (-x 'R, 8R)}.

The structure of the model can be compared with those of previous studies. As far as attrition is concerned, this Chapter assumes that data availability at the start of the transition is exogenous, as in Hausman and Wise [1979] and differently from Bingley et al. [1995], who include also those workers with a valid wage only at the end of the transition in the control group of their retention probit. Moreover, as mentioned above, while Hausman and Wise analyse attrition from the sample of respondents, either Bingley et al. and this Chapter consider attrition from the sample of wage earners. Taking the modelling of wage dynamics into account, Hausman and Wise have no dynamics in their specification (and hence no initial conditions problem), the present Chapter conditions wage levels on their lags, while Bingley et al. condition wage changes on lagged wages. It has to be stressed that the Bingley et al.’s specification and the one in this Chapter are observationally equivalent when mobility from the tails of the wage distribution is considered.

As for the bivariate model, restrictions in the form of variables entering only xR would be desirable. Here I assume that such variables are given by some of the reservation wage indicators included in Table 5.1, namely the dummy for the presence of dependent children in the household interacted with the gender dummy. Such variables have been chosen on the basis of a reduced form bivariate probit model in which Rj and di93 have been conditioned on a general specification of xRi80; results from the reduced form show how these two variables do not enter the low- pay equation significantly81; it is then assumed that their effect on wages in both time periods only works through participation in 1995. The choice of such variables is also in line with previous studies of attrition bias in panel wage analysis, namely with Keane et al. [1988], who use the number of kids as an instrument in their

The specification includes the whole set of variables used in column 2 of Table 5.1 plus the parental background indicators of Chapter 4.

The p-value for these variables in the 1993 low-pay equation is .77 for the male dummy and .9 for the female one.

5. Discontinuous wage profiles, endogenous selection and mobility: a simulated estimation

approach

employment equation. As mentioned in the introduction to the Chapter, attrition may well result from demand side factors, and it could also be argued that such factors are more relevant at the lower end of the wage distribution, where monopsonistic behaviour is likely to characterise the labour market (see Green et al. [1996]). However, it is difficult to imagine demand side factors, among the available information, which do not enter the wage equation directly.

As mentioned in the introduction, we can see from (5.5) that the log-likelihood function involves the cumulative density function of the trivariate normal distribution, whose evaluation has been implemented via simulation estimation: the appendix to this Chapter illustrates the practical implementation of such an estimator.

5. Discontinuous wage profiles, endogenous selection and mobility: a simulated estimation

approach

5.4 Results

Results from the simulated maximum likelihood analysis are given in Table 5.3, which reports SML estimated coefficients and asymptotic t-ratios for the two nesting equations and the low-pay transition equation; the analysis is restricted to the low- pay threshold defined in terms of the bottom quintile of the hourly wage distribution, while, as a benchmark, the first column of the Table reports results obtained with a nested bivariate model which only controls for the endogeneity of initial conditions.82 The simulated likelihood function is computed using 75 random draws from the truncated normal distributions of interest.83

Column 2 reports results from a general specification of the trivariate model. By comparing estimated coefficients with the ones in the first column, we can observe that differences are not remarkable, a fact which suggests that results from the

82 Results in column 1 are taken from Table 7 in Chapter 4, where bivariate and ordered probit